Encyclopedia of Microtonal Music Theory

Werckmeister well-temperaments

Andreas Werckmeister proposed a number of temperaments which fall into the category now known as "well-temperaments", also called "circulating" or "irregular" temperaments.

In contrast to the various meantone tunings, which are based on tempering by fractions of the syntonic comma, Werckmeister's well-temperaments are all based on tempering by fractions of the pythagorean comma.

These tunings are closed 12-tone systems, intended primarily for use on keyboards; thus, all pitches mapped to the black-keys can be taken as either sharps or flats.

The first two tunings which were described by Werckmeister were not considered "good": just-intonation ("Werckmeister I"), which he considered "too perfect", and an extended form of 1/4-comma meantone with more than 12 notes ("Werckmeister II"), where the keyboard had split keys, and which he considered "incorrect". He is known today primarily for his "III" temperament, which is analyzed here in detail. (More information will follow in the future about his other temperaments.)

Werckmeister III: "Correct Temperament No. 1" [-Barbour]

In octave-equivalent terms, here's how Werckmeister III works:

The 4 "5ths" between C:G:D:A and B:F# are tuned 1/4 of a pythagorean comma narrow, and all the rest of the "5ths" are tuned to the Pythagorean 3:2 ratio.

(Note: in this table, fractional exponents of 2 must be given for the Werckmeister 5ths, but since the scale was normally tuned first in one reference octave and then other 8ves tuned from that, any of the exponents of 2 can be adjusted +/- any integer value without affecting the properties inherent in the tuning; thus, the G# [11 -4] tuned a narrow 5th above C# is essentially the same as the G# [4 -4] tuned a 3:2 below Eb -- the exponent of 3 is -4 in both cases, and the 7-octave difference of 2(11-4) = 27 is irrelevant.)

The Werckmeister 5ths are those between C:G:D:A and B:F#/Gb. The 5ths between Gb/F#:Db/C#:Ab/G#:Eb/D#:Bb/A#:F:C and A:E:B are 3:2 ratios.

612-edo gives a superb approximation of Werckmeister III, the maximum deviation being only ~1/29-cent. Below is a graph of Werckmeister III tuning as a scale within one octave, given as 612-edo degrees. The major divisions on the y-axis quantize it into the 612-edo representations of 12-edo for comparison. Note that 612-edo divides exactly into 12, so it provides an excellent means of comparison between 12-edo and Werckmeister III without the need for decimal or fractional parts.

200-edo also gives an excellent approximation of Werckmeister III, the maximum deviation being only ~1/4-cent. Below is a graph of Werckmeister III tuning as a scale within one octave, given as 200-edo degrees. The major divisions on the y-axis quantize it into the 200-edo representations of 12-edo for comparison; note that 200-edo does not divide evenly into 12.

Again, as in the Correct Temperament No. 1 above, the exponents of 2 for G# only signify 8ve-register, and have no effect on 8ve-invariant aspects of the tuning, signified by 3-8. Note also that Barbour's Table 136 (on p 160) has an error: the tuning of G# is given as 786 cents, but should be 784; the table above agrees with Barbour's description of this tuning as "containing 5 5ths flat by 1/3 comma, 2 5ths sharp by 1/3 comma, and 5 perfect 5ths".

The last tuning described by Werckmeister is known as the "Septenarius", because it is based on a string-length of 196 = 22 * 72. As Werckmeister himself wrote, this tuning is different from the three other "good" tunings because it is not based on a tempering by a division of the comma -- instead it is actually an example of an RI (rational-intonation) which is not a JI (just-intonation). In effect it is a well-temperament similar to the others, but created in a different way.